Whitney Extension Theorem - Extension in A Half Space

Extension in A Half Space

Seeley (1964) proved a sharpening of the Whitney extension theorem in the special case of a half space. A smooth function on a half space Rn,+ of points where x_n ≥ 0 is a smooth function f on the interior x_n for which the derivatives ∂α f extend to continuous functions on the half space. On the boundary x_n = 0, f restricts to smooth function. By Borel's lemma can be extended to a smooth function on the whole of Rn. Since Borel's lemms is local in nature, the same argument shows that if Ω is a (bounded or unbounded) domain in Rn with smooth boundary, then any smooth function on the closure of Ω can be extended to a smooth function on Rn.

Seeley's result for a half line gives a uniform extension map

which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in into functions supported in

To define E, set

where φ is a smooth function of compact support on R equal to 1 near 0 and the sequences (a_m), (b_m) satisfy:

• b_m > 0 tends to ∞;
• ∑ a_m b_mj = (−1)j for j ≥ 0 with the sum absolutely convergent.

A solution to this system of equations can be obtained by taking b_n = 2n and seeking an entire function

such that g(2j) = (−1)j. That such a function can be constructed follows from the Weierstrass theorem and Mittag-Leffler theorem.

It can be seen directly by setting

an entire function with simple zeros at 2j. The derivatives W '(2j) are bounded above and below. Similarly the function

meromorphic with simple poles and prescribed residues at 2j.

By construction

is an entire function with the required properties.

The definition for a half space in Rn by applying the operator R to the last variable x_n. Similarly, using a smooth partition of unity and a local change of variables, the result for a half space implies the existence of an analogous extending map

for any domain Ω in Rn with smooth boundary.